Editing Froude number

{{short description|Dimensionless number; ratio of a fluid's flow inertia to the external field}}

In [[continuum mechanics]], the '''Froude number''' ({{math|'''Fr'''}}, after [[William Froude]], {{IPAc-en|ˈ|f|r|uː|d}}<ref>Merriam Webster Online (for brother [[James Anthony Froude]]) [http://www.merriam-webster.com/dictionary/froude]</ref>) is a [[dimensionless number]] defined as the ratio of the [[Viscosity|flow inertia]] to the [[body force|external force field]] (the latter in many applications simply due to [[gravity]]). The Froude number is based on the '''speed–length ratio''' which he defined as:{{sfn|Shih|2009|p=7}}{{sfn|White|1999|p=294}}
<math display="block">\mathrm{Fr} = \frac{u}{\sqrt{g L}}</math>
where {{mvar|u}} is the local [[flow velocity]] (in m/s), {{mvar|g}} is the local [[gravity field]] (in m/s<sup>2</sup>), and {{mvar|L}} is a [[characteristic length]] (in m). 

The Froude number has some analogy with the [[Mach number]]. In theoretical [[fluid dynamics]] the Froude number is not frequently considered since usually the equations are considered in the high Froude limit of negligible external field, leading to homogeneous equations that preserve the mathematical aspects. For example, homogeneous [[Euler equations (fluid dynamics)|Euler equations]] are [[conservation law|conservation equations]].
However, in [[naval architecture]] the Froude number is a significant figure used to determine the resistance of a partially submerged object moving through water.

==Origins==
In [[Open-channel flow|open channel flows]], {{harvnb|Belanger|1828|p=}} introduced first the ratio of the flow velocity to the square root of the gravity acceleration times the flow depth. When the ratio was less than unity, the flow behaved like a fluvial motion (i.e., subcritical flow), and like a torrential flow motion when the ratio was greater than unity.{{sfn|Chanson|2009|pp=159–163}}

[[Image:Boat models by William Froude.JPG|thumb|right|The hulls of ''swan'' (above) and ''raven'' (below). A sequence of 3, 6, and 12 (shown in the picture) foot scale models were constructed by Froude and used in towing trials to establish resistance and scaling laws.]] Quantifying resistance of floating objects is generally credited to [[William Froude]], who used a series of scale models to measure the resistance each model offered when towed at a given speed.  The naval constructor [[Frederic Reech]] had put forward the concept much earlier in 1852 for testing ships and propellers but Froude was unaware of it.{{sfn|Normand|1888|pp=257-261}} Speed–length ratio was originally defined by Froude in his ''Law of Comparison'' in 1868 in dimensional terms as:
<math display="block">\text{speed–length ratio} =\frac{u}{\sqrt {\text{LWL}} }</math>
where:
*{{math|''u''}} = flow speed
*{{math|LWL}} = length of waterline

The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. In France, it is sometimes called '''Reech–Froude number''' after Frederic Reech.{{sfn|Chanson|2004|p= xxvii}}

==Definition and main application==
To show how the Froude number is linked to general continuum mechanics and not only to [[hydrodynamics]] we start from the Cauchy momentum equation in its dimensionless (nondimensional) form.

===Cauchy momentum equation===
{{see also|Cauchy momentum equation}}
In order to make the equations dimensionless, a characteristic length r<sub>0</sub>, and a characteristic velocity u<sub>0</sub>, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
<math display="block"> \rho^*\equiv \frac \rho {\rho_0}, \quad u^*\equiv \frac u {u_0}, \quad r^*\equiv \frac r {r_0}, \quad t^*\equiv \frac {u_0}{r_0} t, \quad \nabla^*\equiv r_0 \nabla , \quad \mathbf g^* \equiv \frac {\mathbf g} {g_0}, \quad \boldsymbol \sigma^* \equiv \frac {\boldsymbol \sigma} {p_0}, </math>

Substitution of these inverse relations in the Euler momentum equations, and definition of the Froude number:
<math display="block">\mathrm{Fr}=\frac{u_0}{\sqrt{g_0 r_0}},</math>
and the [[Euler number (physics)|Euler number]]:
<math display="block">\mathrm{Eu}=\frac{p_0}{\rho_0 u_0^2},</math>
the equations are finally expressed (with the [[material derivative]] and now omitting the indexes):

{{Equation box 1
|indent=:
|title='''Cauchy momentum equation''' (''nondimensional convective form'')
|equation=
<math display="block">
\frac{D \mathbf u}{D t} + \mathrm{Eu} \frac 1 \rho \nabla \cdot \boldsymbol \sigma = \frac 1 {\mathrm{Fr}^2} \mathbf g
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

Cauchy-type equations in the high Froude limit {{math|Fr → ∞}} (corresponding to negligible external field) are named '''free equations'''. On the other hand, in the low Euler limit {{math|Eu → 0}} (corresponding to negligible stress) general Cauchy momentum equation becomes an inhomogeneous '''[[Burgers equation]]''' (here we make explicit the [[material derivative]]):

{{Equation box 1
|indent=:
|title='''Burgers equation''' (''nondimensional conservation form'')
|equation=
<math display="block">
\frac{\partial \mathbf u}{\partial t} + \nabla \cdot \left(\frac 1 2 \mathbf u \otimes \mathbf u \right) = \frac 1 {\mathrm{Fr}^2} \mathbf g
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

This is an inhomogeneous pure [[advection equation]], as much as the [[Stokes flow|Stokes equation]] is a pure [[diffusion equation]].

===Euler momentum equation===
{{see also|Euler equations (fluid dynamics)}}
Euler momentum equation is a Cauchy momentum equation with the [[Pascal law]] being the stress constitutive relation:
<math display="block">\boldsymbol \sigma = p \mathbf I </math>
in nondimensional Lagrangian form is:
<math display="block">\frac{D \mathbf u}{D t} +  \mathrm{Eu} \frac {\nabla p}{\rho}= \frac 1 {\mathrm{Fr}^2} \hat g </math>

Free Euler equations are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with [[perturbation theory]].

===Incompressible Navier–Stokes momentum equation===
{{see also|Navier–Stokes equations#Incompressible flow}}
Incompressible Navier–Stokes momentum equation is a Cauchy momentum equation with the [[Pascal law]] and [[Stokes' law|Stokes's law]] being the stress constitutive relations:
<math display="block">\boldsymbol \sigma = p \mathbf I + \mu \left(\nabla\mathbf{u} +  ( \nabla\mathbf{u} )^\mathsf{T}\right) </math>
in nondimensional convective form it is:{{sfn|Shih|2009|p=}}
<math display="block">\frac{D \mathbf u}{D t} + \mathrm{Eu} \frac {\nabla p}{\rho} = \frac 1 {\mathrm{Re}} \nabla^2 u + \frac 1 {\mathrm{Fr}^2} \hat g </math>
where {{math|Re}} is the [[Reynolds number]]. Free Navier–Stokes equations are [[dissipative system|dissipative]] (non conservative).

==Other applications==

===Ship hydrodynamics===
[[File:Froude numbers and waves.png|thumb|300px|Wave pattern versus speed, illustrating various Froude numbers.]]
In marine hydrodynamic applications, the Froude number is usually referenced with the notation {{math|Fn}} and is defined as:{{sfn |Newman|1977|p=28}}
<math display="block">\mathrm{Fn}_L = \frac{u}{\sqrt{gL}},</math>
where {{math|''u''}} is the relative flow velocity between the sea and ship, {{math|''g''}} is in particular the [[Gravity of Earth|acceleration due to gravity]], and {{math|''L''}} is the length of the ship at the water line level, or {{math|''L''<sub>wl</sub>}} in some notations. It is an important parameter with respect to the ship's [[drag (physics)|drag]], or resistance, especially in terms of [[wave making resistance]].

In the case of planing craft, where the waterline length is too speed-dependent to be meaningful, the Froude number is best defined as ''displacement Froude number'' and the reference length is taken as the cubic root of the volumetric displacement of the hull:
<math display="block">\mathrm{Fn}_V = \frac{u}{\sqrt{g\sqrt[3]{V}}}.</math>

===Shallow water waves===
For shallow water waves, such as [[tsunami]]s and [[hydraulic jump]]s, the characteristic velocity {{math|''U''}} is the [[average]] flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, termed celerity {{math|''c''}}, is equal to the square root of gravitational acceleration {{math|''g''}}, times cross-sectional area {{math|''A''}}, divided by free-surface width {{math|''B''}}:
<math display="block">c = \sqrt{g \frac{A}{B}},</math>
so the Froude number in shallow water is:
<math display="block">\mathrm{Fr} = \frac{U}{\sqrt{g \dfrac{A}{B}}}.</math>
For rectangular cross-sections with uniform depth {{math|''d''}}, the Froude number can be simplified to:
<math display="block">\mathrm{Fr} = \frac{U}{\sqrt{gd}}.</math>
For {{math|Fr < 1}} the flow is called a [[subcritical flow]], further for {{math|Fr > 1}} the flow is characterised as [[supercritical flow]]. When {{math|Fr ≈ 1}} the flow is denoted as '''critical flow'''.

===Wind engineering===
When considering [[Wind engineering|wind effects]] on dynamically sensitive structures such as suspension bridges it is sometimes necessary to simulate the combined effect of the vibrating mass of the structure with the fluctuating force of the wind. In such cases, the Froude number should be respected.  
Similarly, when simulating hot smoke plumes combined with natural wind, Froude number scaling is necessary to maintain the correct balance between buoyancy forces and the momentum of the wind.

=== Allometry ===
The Froude number has also been applied in [[allometry]] to studying the [[Terrestrial locomotion|locomotion]] of terrestrial animals,<ref>{{Cite book |last=Alexander |first=R. McNeill |title=Functional Vertebrate Morphology |chapter-url=https://www.degruyter.com/document/doi/10.4159/harvard.9780674184404.c2/html |chapter=Chapter 2. Body Support, Scaling, and Allometry |date=2013-10-01 |pages=26–37 |publisher=Harvard University Press |isbn=978-0-674-18440-4 |language=en |doi=10.4159/harvard.9780674184404.c2}}</ref> including antelope<ref>{{Cite journal |last=Alexander |first=R. McN. |date=1977 |title=Allometry of the limbs of antelopes (Bovidae) |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1469-7998.1977.tb04177.x |journal=Journal of Zoology |language=en |volume=183 |issue=1 |pages=125–146 |doi=10.1111/j.1469-7998.1977.tb04177.x |issn=0952-8369|url-access=subscription }}</ref> and dinosaurs.<ref>{{Cite journal |last=Alexander |first=R. McNeill |date=1991 |title=How Dinosaurs Ran |url=https://www.jstor.org/stable/24936872 |journal=Scientific American |volume=264 |issue=4 |pages=130–137 |doi=10.1038/scientificamerican0491-130 |jstor=24936872 |bibcode=1991SciAm.264d.130A |issn=0036-8733|url-access=subscription }}</ref>

==Extended Froude number==
Geophysical mass flows such as [[avalanche]]s and [[debris flow]]s take place on inclined slopes which then merge into gentle and flat run-out zones.{{sfn|Takahashi|2007|p=6}}

So, these flows are associated with the elevation of the topographic slopes that induce the gravity potential energy together with the pressure potential energy during the flow. Therefore, the classical Froude number should include this additional effect. For such a situation, Froude number needs to be re-defined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:
<math display="block">\mathrm{Fr} = \frac{u}{\sqrt{\beta h + s_g \left(x_d - x\right)}},</math>
where {{math|''u''}} is the mean flow velocity, {{math|1=''β'' = ''gK'' cos ''ζ''}}, ({{math|''K''}} is the [[Lateral earth pressure|earth pressure coefficient]], {{math|''ζ''}} is the slope), {{math|1=''s<sub>g</sub>'' = ''g'' sin ''ζ''}}, {{math|''x''}} is the channel downslope position and <math>x_d</math> is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; {{math|1=''E''{{su|b=pot|p=''p''}} = ''βh''}} and {{math|1=''E''{{su|b=pot|p=''g''}} = ''s<sub>g</sub>''(''x<sub>d</sub>'' − ''x'')}} are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallow-water or granular flow Froude number, the potential energy associated with the surface elevation, {{math|''E''{{su|b=pot|p=''g''}}}}, is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term {{math|''βh''}} emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows {{math|''βh'' ≪ 1}}, while {{math|''u''}} and {{math|''s<sub>g</sub>''(''x<sub>d</sub>'' − ''x'')}} are both of order unity. If the mass is shallow with a virtually bed-parallel free-surface, then {{math|''βh''}} can be disregarded. In this situation, if the gravity potential is not taken into account, then {{math|Fr}} is unbounded even though the kinetic energy is bounded. So, formally considering the additional contribution due to the gravitational potential energy, the singularity in Fr is removed.

===Stirred tanks===
In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is {{math|''ωr''}} ([[circular motion]]), where {{math|''ω''}} is the impeller frequency (usually in [[Revolutions per minute|rpm]]) and {{math|''r''}} is the impeller radius (in engineering the diameter is much more frequently employed), the Froude number then takes the following form:
<math display="block">\mathrm{Fr}=\omega \sqrt \frac{r}{g}.</math>
The Froude number finds also a similar application in powder mixers. It will indeed be used to determine in which mixing regime the blender is working. If Fr<1, the particles are just stirred, but if Fr>1, centrifugal forces applied to the powder overcome gravity and the bed of particles becomes fluidized, at least in some part of the blender, promoting mixing<ref name="powderprocess.net" />

===Densimetric Froude number===

When used in the context of the [[Boussinesq approximation (buoyancy)|Boussinesq approximation]] the '''densimetric Froude number''' is defined as
<math display="block">\mathrm{Fr}=\frac{u}{\sqrt{g' h}}</math>
where {{math|''g''′}} is the reduced gravity:
<math display="block">g' = g\frac{\rho_1-\rho_2}{\rho_1}</math>

The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the [[Richardson number]] which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a [[gravity current]] moves with a front Froude number of about unity.

===Walking Froude number===

The Froude number may be used to study trends in animal gait patterns. In analyses of the dynamics of legged locomotion, a walking limb is often modeled as an inverted [[pendulum]], where the center of mass goes through a circular arc centered at the foot.{{sfn|Vaughan|O'Malley|2005|pp=350–362}} The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:
<math display="block">\mathrm{Fr}=\frac{\text{centripetal force}}{\text{gravitational force}}=\frac{\;\frac{mv^2}{l}\;}{mg} = \frac{v^2}{gl}</math>
where {{math|''m''}} is the mass, {{math|''l''}} is the characteristic length, {{math|''g''}} is the [[Earth's gravity|acceleration due to gravity]] and {{math|''v''}} is the [[velocity]]. The characteristic length {{math|''l''}} may be chosen to suit the study at hand. For instance, some studies have used the vertical distance of the hip joint from the ground,{{sfn|Alexander|1984|p=}} while others have used total leg length.{{sfn|Vaughan|O'Malley|2005|pp=350–362}}{{sfn|Sellers|Manning|2007|p=}}

The Froude number may also be calculated from the stride frequency {{math|''f''}} as follows:{{sfn|Alexander|1984|p=}}
<math display="block">\mathrm{Fr}=\frac{v^2}{gl}=\frac{(lf)^2}{gl}=\frac{lf^2}{g}.</math>

If total leg length is used as the characteristic length, then the theoretical maximum speed of walking has a Froude number of 1.0 since any higher value would result in takeoff and the foot missing the ground. The typical transition speed from bipedal walking to [[running]] occurs with {{math|Fr ≈ 0.5}}.{{sfn|Alexander| 1989|p=}}  R. M. Alexander found that animals of different sizes and masses travelling at different speeds, but with the same Froude number, consistently exhibit similar gaits. This study found that animals typically switch from an amble to a symmetric running gait (e.g., a trot or pace) around a Froude number of 1.0. A preference for asymmetric gaits (e.g., a canter, transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between 2.0 and 3.0.{{sfn|Alexander|1984|p=}}

==Usage==

The Froude number is used to compare the [[wave making resistance]] between bodies of various sizes and shapes.

In free-surface flow, the nature of the flow ([[Supercritical flow|supercritical]] or subcritical) depends upon whether the Froude number is greater than or less than unity.

One can easily see the line of "critical" flow in a kitchen or bathroom sink. Leave it unplugged and let the faucet run. Near the place where the stream of water hits the sink, the flow is supercritical. It 'hugs' the surface and moves quickly. On the outer edge of the flow pattern the flow is subcritical. This flow is thicker and moves more slowly. The boundary between the two areas is called a "hydraulic jump". The jump starts where the flow is just critical and Froude number is equal to 1.0.

The Froude number has been used to study trends in animal locomotion in order to better understand why animals use different gait patterns{{sfn|Alexander|1984|p=}} as well as to form hypotheses about the gaits of extinct species.{{sfn|Sellers|Manning|2007|p=}}

In addition particle bed behavior can be quantified by Froude number (Fr) in order to establish the optimum operating window.{{sfn | Jikar | Dhokey | Shinde|2021 | p=}}

==See also==
* {{annotated link|Flow velocity}}
* {{annotated link|Body force}}
* {{annotated link|Cauchy momentum equation}}
* {{annotated link|Burgers' equation}}
* {{annotated link|Euler equations (fluid dynamics)}}
* {{annotated link|Reynolds number}}

== Notes ==
{{Reflist|colwidth=35em|refs=
<ref name="powderprocess.net">{{Cite web |title=Powder Mixing - Powder Mixers Design - Ribbon blender, Paddle mixer, Drum blender, Froude Number |work=powderprocess.net |date=n.d. |access-date=31 May 2019 |url= https://www.powderprocess.net/Mixing.html }}</ref>
}}

== References ==
{{refbegin|2|indent=yes}}
*{{cite journal |doi=10.1177/027836498400300205 |title=The Gaits of Bipedal and Quadrupedal Animals |year=1984 |last1=Alexander |first1=R. McN. |journal=The International Journal of Robotics Research |volume=3 |issue=2 |pages=49–59 |s2cid=120138903 |doi-access= }}
*{{cite journal |pmid=2678167 |year=1989 |last1=Alexander |first1=RM |title=Optimization and gaits in the locomotion of vertebrates |volume=69 |issue=4 |pages=1199–227 |journal=Physiological Reviews|doi=10.1152/physrev.1989.69.4.1199}}
*{{cite book|last=Belanger|first=Jean Baptiste |author-link=Jean-Baptiste Bélanger |title=Essai sur la solution numerique de quelques problemes relatifs au mouvement permanent des eaux courantes|url=https://books.google.com/books?id=AZ111SlLXbcC&pg=PA1|year=1828|publisher=Carilian-Goeury|location=Paris|lang=fr|trans-title=An essay on the numerical solution to some problems relative to the steady movement of running water}}
*{{Cite book | title=Hydraulics of Open Channel Flow: An Introduction  | first=Hubert | last=Chanson | author-link=Hubert Chanson | publisher=Butterworth–Heinemann | year=2004 | isbn=978-0-7506-5978-9 | edition=2nd | url=https://books.google.com/books?id=VCNmKQI6GiEC|pages=650}}
*{{cite journal |doi=10.1061/(ASCE)0733-9429(2009)135:3(159) |title=Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828) |year=2009 |last1=Chanson |first1=Hubert |journal=Journal of Hydraulic Engineering |volume=135 |issue=3 |pages=159–63|url=http://espace.library.uq.edu.au/view/UQ:165002/jhe09_03.pdf }}
* {{cite journal | last1=Jikar | first1=P. C. | last2=Dhokey | first2=N. B. | last3=Shinde | first3=S. S. | title=Numerical Modeling Simulation and Experimental Study of Dynamic Particle Bed Counter Current Reactor and Its Effect on Solid–Gas Reduction Reaction | journal=Mining, Metallurgy & Exploration | publisher=Springer  | date=2021 | volume=39 | pages=139–152 | issn=2524-3462 | doi=10.1007/s42461-021-00516-6| s2cid=244507908 }}
*{{Cite book | last=Newman | first=John Nicholas | author-link=John Nicholas Newman | title=Marine hydrodynamics | year=1977 | publisher=[[MIT Press]] | location=Cambridge, Massachusetts  | isbn=978-0-262-14026-3 }}
*{{cite journal|title=On the Fineness of vessels in relation to size and speed| last=Normand|first= J.A.|journal=Transactions of the Institution of Naval Architects|volume=29|pages=257–261 |year=1888}}
*{{cite journal |doi=10.1098/rspb.2007.0846 |title=Estimating dinosaur maximum running speeds using evolutionary robotics |year=2007 |last1=Sellers |first1=William Irvin |last2=Manning |first2=Phillip Lars |journal=Proceedings of the Royal Society B: Biological Sciences |volume=274 |issue=1626 |pages=2711–6 |jstor=25249388 |pmid=17711833 |pmc=2279215}}
*{{citation|title=Fluid Mechanics |first=Y.C. |last=Shih |date=Spring 2009|chapter=Chapter 6 Incompressible Inviscid Flow|chapter-url=https://erac.ntut.edu.tw/ezfiles/39/1039/img/832/Ch6-IncompressibleInviscidFlow.pdf}}
*{{cite book|last=Takahashi|first=Tamotsu |title=Debris Flow: Mechanics, Prediction and Countermeasures|url=https://books.google.com/books?id=rh1DuzNJqHAC|year=2007|publisher=CRC Press|isbn=978-0-203-94628-2}}
*{{cite journal |doi=10.1016/j.gaitpost.2004.01.011 |title=Froude and the contribution of naval architecture to our understanding of bipedal locomotion |year=2005 |last1=Vaughan |first1=Christopher L. |last2=O'Malley |first2=Mark J. |journal=Gait & Posture |volume=21 |issue=3 |pages=350–62 |pmid=15760752}}
*{{cite book|last=White|first=Frank M. |title=Fluid mechanics|url=https://books.google.com/books?id=fa_pAAAAMAAJ|edition=4th|year=1999|publisher=WCB/McGraw-Hill|isbn=978-0-07-116848-9}}
{{refend}}

==External links==
* https://web.archive.org/web/20070927085042/http://www.qub.ac.uk/waves/fastferry/reference/MCA457.pdf

{{NonDimFluMech}}

[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Fluid dynamics]]
[[Category:Naval architecture]]